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Note that both Xv and Dv (through the jump rate) are highly temperature dependent. 2 completed. While we have not specified each step, we have the input (lattice and jump rate) and output (diffusion coefficient), we have identified the mechanisms (random jumps), we have targeted the precision (since we cannot predict kj ump we cannot predict a value for D so these will be qualitative, not quantitative, results), we have constructed the model (the sequences of random jumps), and we have done a dimensional analysis (D must be independent of time).

Xn = n a), which is, of course, impossible. However, in any practical sense, the Gaussian distribution is “exact”. 14) where Rn = (xn , yn , zn ) and with expressions similar to Eq. 13) for I (yn ) and I (zn ). P (Rn ) gives the probability that the vector Rn is at a position (xn , yn , zn ). , a measure of how far the atom has diffused in n steps. Since Rn is a vector, P (Rn ) includes angular information that is not needed to determine an end-to-end distance. We need to average out the angles, which we can do by changing (x, y, z) in P (Rn ) to spherical polar coordinates and integrating over the angles.

Value of the peak probability by finding the maximum value of the function, which we do by max taking the derivative of P (Rn ) with respect to Rn , setting it equal to zero, and solving for Rm , √ max the distance of the peak probability. In this case, Rn = 2 n/3a. 4. 17) which is exactly what we determined in Eq. 9). 3 BULK DIFFUSION ............................................................................................. We have developed a simple model for diffusion on an empty lattice, which is not really a problem that is very interesting from a materials perspective.